Beta-binomial distribution

In probability theory and statistics, the beta-binomial distribution is a family of discrete probability distributions on a finite support of non-negative integers arising when the probability of success in each of a fixed or known number of Bernoulli trials is either unknown or random. The beta-binomial distribution is the binomial distribution in which the probability of success at each of n trials is not fixed but randomly drawn from a beta distribution. It is frequently used in Bayesian statistics, empirical Bayes methods and classical statistics to capture overdispersion in binomial type distributed data.

Probability mass function
Cumulative distribution function
Notation	${\displaystyle \mathrm {BetaBin} (n,\alpha ,\beta )}$
Parameters	n ∈ N0 — number of trials ${\displaystyle \alpha >0}$ (real) ${\displaystyle \beta >0}$ (real)
Support	x ∈ { 0, …, n }
PMF	${\displaystyle {\binom {n}{x}}{\frac {\mathrm {B} (x+\alpha ,n-x+\beta )}{\mathrm {B} (\alpha ,\beta )}}\!}$ where ${\displaystyle \mathrm {B} (x,y)={\frac {\Gamma (x)\,\Gamma (y)}{\Gamma (x+y)}}}$ is the beta function
CDF	${\displaystyle {\begin{cases}0,&x<0\\{\binom {n}{x}}{\tfrac {\mathrm {B} (x+\alpha ,n-x+\beta )}{\mathrm {B} (\alpha ,\beta )}}{}_{3}\!F_{2}({\boldsymbol {a}};{\boldsymbol {b}};x),&0\leq x<n\\1,&x\geq n\end{cases}}}$ where 3F2(a;b;x) is the generalized hypergeometric function ${\displaystyle {}_{3}\!F_{2}(1,-x,n\!-\!x\!+\!\beta ;n\!-\!x\!+\!1,1\!-\!x\!-\!\alpha ;1)\!}$
Mean	${\displaystyle {\frac {n\alpha }{\alpha +\beta }}\!}$
Variance	${\displaystyle {\frac {n\alpha \beta (\alpha +\beta +n)}{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!}$
Skewness	${\displaystyle {\tfrac {(\alpha +\beta +2n)(\beta -\alpha )}{(\alpha +\beta +2)}}{\sqrt {\tfrac {1+\alpha +\beta }{n\alpha \beta (n+\alpha +\beta )}}}\!}$
Excess kurtosis	See text
MGF	${\displaystyle _{2}F_{1}(-n,\alpha$ ;\alpha +\beta ;1-e^{t})\!} where ${\displaystyle _{2}F_{1}}$ is the hypergeometric function
CF	${\displaystyle _{2}F_{1}(-n,\alpha$ ;\alpha +\beta ;1-e^{it})\!}
PGF	${\displaystyle _{2}F_{1}(-n,\alpha$ ;\alpha +\beta ;1-z)\!}

The beta-binomial is a one-dimensional version of the Dirichlet-multinomial distribution as the binomial and beta distributions are univariate versions of the multinomial and Dirichlet distributions respectively. The special case where α and β are integers is also known as the negative hypergeometric distribution.